Properties

Label 4014.2.a.u
Level 4014
Weight 2
Character orbit 4014.a
Self dual Yes
Analytic conductor 32.052
Analytic rank 0
Dimension 6
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4014.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0519513713\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.103354048.1
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + q^{4} + ( -\beta_{4} + \beta_{5} ) q^{5} + ( 1 - \beta_{4} ) q^{7} + q^{8} +O(q^{10})\) \( q + q^{2} + q^{4} + ( -\beta_{4} + \beta_{5} ) q^{5} + ( 1 - \beta_{4} ) q^{7} + q^{8} + ( -\beta_{4} + \beta_{5} ) q^{10} + ( \beta_{3} - \beta_{4} ) q^{11} + ( -\beta_{1} + \beta_{5} ) q^{13} + ( 1 - \beta_{4} ) q^{14} + q^{16} + ( \beta_{2} - \beta_{3} ) q^{17} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{19} + ( -\beta_{4} + \beta_{5} ) q^{20} + ( \beta_{3} - \beta_{4} ) q^{22} + ( 4 - \beta_{1} ) q^{23} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} + ( -\beta_{1} + \beta_{5} ) q^{26} + ( 1 - \beta_{4} ) q^{28} + ( 3 - \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{29} + ( -2 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{31} + q^{32} + ( \beta_{2} - \beta_{3} ) q^{34} + ( 3 + \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{35} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{5} ) q^{37} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{38} + ( -\beta_{4} + \beta_{5} ) q^{40} + ( 4 + \beta_{1} + \beta_{2} ) q^{41} + ( 3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{43} + ( \beta_{3} - \beta_{4} ) q^{44} + ( 4 - \beta_{1} ) q^{46} + ( 2 - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{47} + ( -1 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{49} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{50} + ( -\beta_{1} + \beta_{5} ) q^{52} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{53} + ( 1 + 2 \beta_{1} - \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{55} + ( 1 - \beta_{4} ) q^{56} + ( 3 - \beta_{1} - 2 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{58} + ( 1 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{59} + ( -4 \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{61} + ( -2 - \beta_{1} + 2 \beta_{3} - \beta_{4} + \beta_{5} ) q^{62} + q^{64} + ( 4 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{5} ) q^{65} + ( 4 - \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{67} + ( \beta_{2} - \beta_{3} ) q^{68} + ( 3 + \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{70} + ( 3 \beta_{2} - 3 \beta_{4} + \beta_{5} ) q^{71} + ( -3 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{5} ) q^{73} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{5} ) q^{74} + ( -1 + \beta_{2} - \beta_{3} - \beta_{4} ) q^{76} + ( 4 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{77} + ( 3 - 3 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} ) q^{79} + ( -\beta_{4} + \beta_{5} ) q^{80} + ( 4 + \beta_{1} + \beta_{2} ) q^{82} + ( 2 - \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{83} + ( -2 + \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{85} + ( 3 + 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{86} + ( \beta_{3} - \beta_{4} ) q^{88} + ( 2 - 6 \beta_{1} - \beta_{2} - \beta_{5} ) q^{89} + ( -2 - 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{91} + ( 4 - \beta_{1} ) q^{92} + ( 2 - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{94} + ( 1 + 2 \beta_{1} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{95} + ( -3 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{97} + ( -1 + 2 \beta_{1} - 2 \beta_{3} - \beta_{4} - \beta_{5} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{2} + 6q^{4} + 2q^{5} + 8q^{7} + 6q^{8} + O(q^{10}) \) \( 6q + 6q^{2} + 6q^{4} + 2q^{5} + 8q^{7} + 6q^{8} + 2q^{10} + 2q^{11} - 2q^{13} + 8q^{14} + 6q^{16} + 2q^{17} - 2q^{19} + 2q^{20} + 2q^{22} + 22q^{23} + 12q^{25} - 2q^{26} + 8q^{28} + 8q^{29} - 12q^{31} + 6q^{32} + 2q^{34} + 20q^{35} - 2q^{38} + 2q^{40} + 28q^{41} + 14q^{43} + 2q^{44} + 22q^{46} + 2q^{47} + 12q^{50} - 2q^{52} + 26q^{53} + 6q^{55} + 8q^{56} + 8q^{58} + 4q^{59} - 6q^{61} - 12q^{62} + 6q^{64} + 16q^{65} + 18q^{67} + 2q^{68} + 20q^{70} + 12q^{71} - 20q^{73} - 2q^{76} + 20q^{77} + 12q^{79} + 2q^{80} + 28q^{82} + 12q^{83} - 10q^{85} + 14q^{86} + 2q^{88} - 2q^{89} - 22q^{91} + 22q^{92} + 2q^{94} + 6q^{95} - 6q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 2 x^{5} - 8 x^{4} + 14 x^{3} + 13 x^{2} - 16 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} - \nu^{2} - 5 \nu + 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} + \nu^{3} - 7 \nu^{2} - 6 \nu + 5 \)
\(\beta_{4}\)\(=\)\( \nu^{5} - \nu^{4} - 8 \nu^{3} + 6 \nu^{2} + 13 \nu - 5 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - \nu^{4} - 8 \nu^{3} + 7 \nu^{2} + 13 \nu - 8 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{4} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{5} - \beta_{4} + \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(6 \beta_{5} - 6 \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1} + 16\)
\(\nu^{5}\)\(=\)\(8 \beta_{5} - 7 \beta_{4} + \beta_{3} + 7 \beta_{2} + 28 \beta_{1} + 3\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0.249769
0.605879
−1.40037
2.34874
−2.35901
2.55499
1.00000 0 1.00000 −2.93762 0 2.50626 1.00000 0 −2.93762
1.2 1.00000 0 1.00000 −2.63291 0 −2.24656 1.00000 0 −2.63291
1.3 1.00000 0 1.00000 −1.03898 0 −0.299718 1.00000 0 −1.03898
1.4 1.00000 0 1.00000 2.51659 0 4.97725 1.00000 0 2.51659
1.5 1.00000 0 1.00000 2.56494 0 2.27923 1.00000 0 2.56494
1.6 1.00000 0 1.00000 3.52797 0 0.783532 1.00000 0 3.52797
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(223\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4014))\):

\( T_{5}^{6} - 2 T_{5}^{5} - 19 T_{5}^{4} + 30 T_{5}^{3} + 110 T_{5}^{2} - 112 T_{5} - 183 \)
\( T_{7}^{6} - 8 T_{7}^{5} + 11 T_{7}^{4} + 36 T_{7}^{3} - 84 T_{7}^{2} + 22 T_{7} + 15 \)
\( T_{11}^{6} - 2 T_{11}^{5} - 22 T_{11}^{4} + 28 T_{11}^{3} + 75 T_{11}^{2} + 42 T_{11} + 7 \)